\section{Coordinate systems}



\subsection{Cylindrical}



\subsubsection*{Basis vectors}
\input{content/figure/cyl}
$\vec{e}_{\rho}=\abs{\pdv{\vec{r}}{\rho}}^{-1}\pdv{\vec{r}}{\rho}=\qty(\cos\varphi,\sin\varphi,0)$

$\vec{e}_{\varphi}=\abs{\pdv{\vec{r}}{\varphi}}^{-1}\pdv{\vec{r}}{\varphi}=\qty(\sin\varphi,-\cos\varphi,0)$

$\vec{e}_{z}=\abs{\pdv{\vec{r}}{z}}^{-1}\pdv{\vec{r}}{z}=\qty(0,0,1)$
\subsubsection*{Velocity vectors}
$\dot{\vec{e}}_{\rho}=\dot{\varphi}\vec{e}_{\varphi},\hsma \dot{\vec{e}}_{\varphi}=-\dot{\varphi}\vec{e}_{\rho},\hsma \dot{\vec{e}}_{z}=0,$

$v_{\rho}=\dot{\rho},\hsma v_{\varphi}=\rho\dot{\varphi},\hsma v_{z}=\dot{z}$.                
\subsubsection*{Acceleration vectors}
$a_{\rho}=\ddot{\rho}-\rho\dot{\varphi}^{2},\hsma a_{\varphi}=\rho\ddot{\varphi}+2\dot{\rho}\dot{\varphi},\hsma a_{z}=\ddot{z}.$

\input{content/figure/sph}
\subsection{Spherical}
\subsubsection*{Basis vectors}
$\vec{e}_{r}=\abs{\pdv{\vec{r}}{r}}^{-1}\pdv{\vec{r}}{r}$

$=\qty(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta)=\frac{\vec{r}}{r}$,

$\vec{e}_{\theta}=\abs{\pdv{\vec{r}}{\theta}}^{-1}\pdv{\vec{r}}{\theta}$

$=\qty(\cos\theta\cos\varphi,\cos\theta\sin\varphi,-\sin\theta)$

$\vec{e}_{\varphi}=\abs{\pdv{\vec{r}}{\varphi}}^{-1}\pdv{\vec{r}}{\varphi}=\qty(\sin\varphi,\cos\varphi,0).$
\subsubsection*{Velocity vectors}
$\dot{\vec{e}}_{r}=\pdv{\vec{e}_{r}}{\theta}\dot{\theta}+\pdv{\vec{e}_{r}}{\varphi}\dot{\varphi}=\dot\theta\vec{e}_{\theta}+\sin\theta\dot\theta\vec{e}_{\varphi},$

$\dot{\vec{e}}_{\theta}=\pdv{\vec{e}_{\theta}}{\theta}\dot{\theta}+\pdv{\vec{e}_{\theta}}{\varphi}\dot{\varphi}=-\dot\theta\vec{e}_{r}+\cos\theta\dot\varphi\vec{e}_{\varphi},$

$\dot{\vec{e}}_{\varphi}=\pdv{\vec{e}_{\varphi}}{\varphi}\dot\varphi=-\qty(\cos\varphi,\sin\varphi,0)\dot\varphi=-\dot\varphi\qty(\sin\theta\vec{e}_{r}+\cos\theta\vec{e}_{\theta}).$

$v_{r}=\dot{r},\hsma v_{\theta}=r\dot{\theta},\hsma v_{\varphi}=r\sin\theta\dot{\varphi}.$
\subsubsection*{Acceleration vectors}
$a_{r}=\ddot{r}-r\dot{\theta}^{2}-r\sin^{2}\theta\dot{\varphi}^{2},\hsma a_{\theta}=r\ddot{\theta}+2\dot{r}\dot{\theta}-r\sin\theta\cos\theta\dot{\varphi}^{2},$

$a_{\varphi}=r\sin\theta\ddot{\varphi}+2\sin\theta\dot{r}\dot{\varphi}+2r\cos\theta\dot{\theta}\dot{\varphi}.$



































